L.C.M of `2/3, 4/9, 5/6`and `7/(12)` is-

Options

A. `1/(18)`
B. `1/(36)`
C. `(35)/9`
D. `(140)/3`

Answer

To find the L.C.M. (Least Common Multiple) of the fractions \( \frac{2}{3}, \frac{4}{9}, \frac{5}{6}, \frac{7}{12} \), we can use the formula:

\[ \text{L.C.M. of fractions} = \frac{\text{L.C.M. of numerators}}{\text{H.C.F. of denominators}} \]

### Step 1: Identify the numerators and denominators
- Numerators: \( 2, 4, 5, 7 \)
- Denominators: \( 3, 9, 6, 12 \)

### Step 2: Calculate the L.C.M. of the numerators
To find the L.C.M. of \( 2, 4, 5, 7 \):
1. The prime factorization of each number:
- \( 2 = 2^1 \)
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
- \( 7 = 7^1 \)

2. Take the highest power of each prime:
- \( 2^2 \) (from 4)
- \( 5^1 \) (from 5)
- \( 7^1 \) (from 7)

3. L.C.M. calculation:
\[ \text{L.C.M.} = 2^2 \times 5^1 \times 7^1 = 4 \times 5 \times 7 \]
\[ 4 \times 5 = 20 \]
\[ 20 \times 7 = 140 \]

So, the L.C.M. of the numerators is \( 140 \).

### Step 3: Calculate the H.C.F. of the denominators
To find the H.C.F. of \( 3, 9, 6, 12 \):
1. The prime factorization of each number:
- \( 3 = 3^1 \)
- \( 9 = 3^2 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 12 = 2^2 \times 3^1 \)

2. Take the lowest power of each common prime:
- \( 3^1 \) (common in all)

So, the H.C.F. of the denominators is \( 3 \).

### Step 4: Calculate the L.C.M. of the fractions
Now we can find the L.C.M. of the fractions:
\[ \text{L.C.M.} = \frac{\text{L.C.M. of numerators}}{\text{H.C.F. of denominators}} = \frac{140}{3} \]

### Final Answer
Thus, the L.C.M. of \( \frac{2}{3}, \frac{4}{9}, \frac{5}{6}, \frac{7}{12} \) is \( \frac{140}{3} \).

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